In The Exercise Below, The Initial Substitution Of $x = A$ Yields The Form $0 / 0$. Look For A Way To Evaluate The Limit:$\lim _{x \rightarrow 2} \frac{x^2+7x-18}{x^2-4}$

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Introduction

In mathematics, limits are a fundamental concept used to study the behavior of functions as the input values approach a specific point. However, when dealing with limits, we often encounter indeterminate forms, which can be challenging to evaluate. In this article, we will explore a step-by-step approach to evaluate the limit of a function that yields an indeterminate form of 0/0.

The Problem

The given problem is to evaluate the limit of the function:

limโกxโ†’2x2+7xโˆ’18x2โˆ’4\lim _{x \rightarrow 2} \frac{x^2+7x-18}{x^2-4}

As we can see, the initial substitution of x=2x = 2 yields the form 0/0, which is an indeterminate form. To evaluate this limit, we need to find a way to simplify the function and avoid the indeterminate form.

Step 1: Factor the Numerator and Denominator

One way to simplify the function is to factor the numerator and denominator. Let's start by factoring the numerator:

x2+7xโˆ’18=(x+9)(xโˆ’2)x^2 + 7x - 18 = (x + 9)(x - 2)

Now, let's factor the denominator:

x2โˆ’4=(x+2)(xโˆ’2)x^2 - 4 = (x + 2)(x - 2)

Step 2: Cancel Common Factors

Now that we have factored the numerator and denominator, we can cancel common factors. In this case, we can cancel the factor (xโˆ’2)(x - 2) from both the numerator and denominator:

(x+9)(xโˆ’2)(x+2)(xโˆ’2)=x+9x+2\frac{(x + 9)(x - 2)}{(x + 2)(x - 2)} = \frac{x + 9}{x + 2}

Step 3: Evaluate the Limit

Now that we have simplified the function, we can evaluate the limit. As xx approaches 2, the function approaches:

limโกxโ†’2x+9x+2=2+92+2=114\lim _{x \rightarrow 2} \frac{x + 9}{x + 2} = \frac{2 + 9}{2 + 2} = \frac{11}{4}

Conclusion

In this article, we have demonstrated a step-by-step approach to evaluate the limit of a function that yields an indeterminate form of 0/0. By factoring the numerator and denominator, canceling common factors, and evaluating the limit, we were able to simplify the function and avoid the indeterminate form. This approach can be applied to a wide range of problems involving limits and indeterminate forms.

Common Indeterminate Forms

In mathematics, there are several common indeterminate forms that can arise when evaluating limits. These include:

  • 0/0
  • โˆž/โˆž
  • 0 ร— โˆž
  • โˆž - โˆž

Each of these forms requires a different approach to evaluate the limit. In the next section, we will explore some common techniques for evaluating limits with these indeterminate forms.

Techniques for Evaluating Limits with Indeterminate Forms

0/0 Form

When dealing with the 0/0 form, we can use the following techniques:

  • L'Hopital's Rule: This rule states that if a limit approaches 0/0, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.
  • Factoring: We can try to factor the numerator and denominator to cancel common factors.
  • Substitution: We can try to substitute a value for the variable that makes the expression finite.

โˆž/โˆž Form

When dealing with the โˆž/โˆž form, we can use the following techniques:

  • L'Hopital's Rule: This rule states that if a limit approaches โˆž/โˆž, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.
  • Factoring: We can try to factor the numerator and denominator to cancel common factors.
  • Substitution: We can try to substitute a value for the variable that makes the expression finite.

0 ร— โˆž Form

When dealing with the 0 ร— โˆž form, we can use the following techniques:

  • L'Hopital's Rule: This rule states that if a limit approaches 0 ร— โˆž, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.
  • Factoring: We can try to factor the numerator and denominator to cancel common factors.
  • Substitution: We can try to substitute a value for the variable that makes the expression finite.

โˆž - โˆž Form

When dealing with the โˆž - โˆž form, we can use the following techniques:

  • L'Hopital's Rule: This rule states that if a limit approaches โˆž - โˆž, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.
  • Factoring: We can try to factor the numerator and denominator to cancel common factors.
  • Substitution: We can try to substitute a value for the variable that makes the expression finite.

Real-World Applications

Limits and indeterminate forms have numerous real-world applications in fields such as physics, engineering, and economics. For example:

  • Physics: Limits are used to study the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
  • Engineering: Limits are used to design and optimize systems, such as electronic circuits or mechanical systems.
  • Economics: Limits are used to study the behavior of economic systems, such as the behavior of supply and demand curves.

Conclusion

Introduction

In our previous article, we explored a step-by-step approach to evaluate the limit of a function that yields an indeterminate form of 0/0. In this article, we will answer some frequently asked questions about evaluating limits with indeterminate forms.

Q: What is an indeterminate form?

A: An indeterminate form is a type of limit that cannot be evaluated using standard limit properties. Indeterminate forms arise when the limit of a function approaches a value that is not defined, such as 0/0, โˆž/โˆž, 0 ร— โˆž, or โˆž - โˆž.

Q: How do I know if a limit is an indeterminate form?

A: To determine if a limit is an indeterminate form, try substituting the value of the variable into the function. If the resulting expression is undefined, then the limit is an indeterminate form.

Q: What are some common techniques for evaluating limits with indeterminate forms?

A: Some common techniques for evaluating limits with indeterminate forms include:

  • L'Hopital's Rule: This rule states that if a limit approaches 0/0 or โˆž/โˆž, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.
  • Factoring: We can try to factor the numerator and denominator to cancel common factors.
  • Substitution: We can try to substitute a value for the variable that makes the expression finite.

Q: What is L'Hopital's Rule?

A: L'Hopital's Rule is a mathematical technique used to evaluate limits that approach 0/0 or โˆž/โˆž. The rule states that if a limit approaches 0/0 or โˆž/โˆž, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.

Q: How do I apply L'Hopital's Rule?

A: To apply L'Hopital's Rule, follow these steps:

  1. Check if the limit approaches 0/0 or โˆž/โˆž.
  2. Take the derivative of the numerator and denominator.
  3. Evaluate the limit of the resulting expression.

Q: What are some common mistakes to avoid when evaluating limits with indeterminate forms?

A: Some common mistakes to avoid when evaluating limits with indeterminate forms include:

  • Not checking if the limit is an indeterminate form: Make sure to check if the limit is an indeterminate form before trying to evaluate it.
  • Not applying L'Hopital's Rule correctly: Make sure to apply L'Hopital's Rule correctly by taking the derivative of the numerator and denominator.
  • Not checking for other techniques: Make sure to check for other techniques, such as factoring or substitution, before applying L'Hopital's Rule.

Q: How do I know if I have evaluated a limit correctly?

A: To determine if you have evaluated a limit correctly, follow these steps:

  1. Check if the limit is an indeterminate form.
  2. Apply the correct technique, such as L'Hopital's Rule or factoring.
  3. Evaluate the limit using the correct technique.
  4. Check if the resulting expression is finite.

Conclusion

In conclusion, evaluating limits with indeterminate forms can be challenging, but with the right techniques and knowledge, it can be done. By understanding how to apply L'Hopital's Rule and other techniques, you can evaluate limits with confidence. Remember to check if the limit is an indeterminate form, apply the correct technique, and evaluate the limit using the correct technique.

Common Indeterminate Forms and Their Solutions

Indeterminate Form Solution
0/0 L'Hopital's Rule
โˆž/โˆž L'Hopital's Rule
0 ร— โˆž L'Hopital's Rule or factoring
โˆž - โˆž L'Hopital's Rule or factoring

Real-World Applications

Limits and indeterminate forms have numerous real-world applications in fields such as physics, engineering, and economics. For example:

  • Physics: Limits are used to study the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
  • Engineering: Limits are used to design and optimize systems, such as electronic circuits or mechanical systems.
  • Economics: Limits are used to study the behavior of economic systems, such as the behavior of supply and demand curves.

Conclusion

In conclusion, limits and indeterminate forms are fundamental concepts in mathematics that have numerous real-world applications. By understanding how to evaluate limits with indeterminate forms, you can gain a deeper understanding of the behavior of functions and systems. This knowledge can be applied to a wide range of problems in physics, engineering, and economics.